Integrand size = 18, antiderivative size = 107 \[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {12 i \sqrt {\cosh (a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right ) \sqrt {\text {sech}(a+b x)}}{35 b^2}-\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {12 \sqrt {\text {sech}(a+b x)} \sinh (a+b x)}{35 b^2}+\frac {4 \text {sech}^{\frac {5}{2}}(a+b x) \sinh (a+b x)}{35 b^2} \]
-2/7*x*sech(b*x+a)^(7/2)/b+4/35*sech(b*x+a)^(5/2)*sinh(b*x+a)/b^2+12/35*si nh(b*x+a)*sech(b*x+a)^(1/2)/b^2+12/35*I*(cosh(1/2*a+1/2*b*x)^2)^(1/2)/cosh (1/2*a+1/2*b*x)*EllipticE(I*sinh(1/2*a+1/2*b*x),2^(1/2))*cosh(b*x+a)^(1/2) *sech(b*x+a)^(1/2)/b^2
Time = 0.42 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.64 \[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\frac {\text {sech}^{\frac {7}{2}}(a+b x) \left (-20 b x+24 i \cosh ^{\frac {7}{2}}(a+b x) E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )+10 \sinh (2 (a+b x))+3 \sinh (4 (a+b x))\right )}{70 b^2} \]
(Sech[a + b*x]^(7/2)*(-20*b*x + (24*I)*Cosh[a + b*x]^(7/2)*EllipticE[(I/2) *(a + b*x), 2] + 10*Sinh[2*(a + b*x)] + 3*Sinh[4*(a + b*x)]))/(70*b^2)
Time = 0.52 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5967, 3042, 4255, 3042, 4255, 3042, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x \sinh (a+b x) \text {sech}^{\frac {9}{2}}(a+b x) \, dx\) |
\(\Big \downarrow \) 5967 |
\(\displaystyle \frac {2 \int \text {sech}^{\frac {7}{2}}(a+b x)dx}{7 b}-\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \int \csc \left (i a+i b x+\frac {\pi }{2}\right )^{7/2}dx}{7 b}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {2 \left (\frac {3}{5} \int \text {sech}^{\frac {3}{2}}(a+b x)dx+\frac {2 \sinh (a+b x) \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}\right )}{7 b}-\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (\frac {2 \sinh (a+b x) \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \int \csc \left (i a+i b x+\frac {\pi }{2}\right )^{3/2}dx\right )}{7 b}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\int \frac {1}{\sqrt {\text {sech}(a+b x)}}dx\right )+\frac {2 \sinh (a+b x) \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}\right )}{7 b}-\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (\frac {2 \sinh (a+b x) \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \left (\frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\int \frac {1}{\sqrt {\csc \left (i a+i b x+\frac {\pi }{2}\right )}}dx\right )\right )}{7 b}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {2 \left (\frac {3}{5} \left (\frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\cosh (a+b x)}dx\right )+\frac {2 \sinh (a+b x) \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}\right )}{7 b}-\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (\frac {2 \sinh (a+b x) \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \left (\frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}-\sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \int \sqrt {\sin \left (i a+i b x+\frac {\pi }{2}\right )}dx\right )\right )}{7 b}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle -\frac {2 x \text {sech}^{\frac {7}{2}}(a+b x)}{7 b}+\frac {2 \left (\frac {2 \sinh (a+b x) \text {sech}^{\frac {5}{2}}(a+b x)}{5 b}+\frac {3}{5} \left (\frac {2 \sinh (a+b x) \sqrt {\text {sech}(a+b x)}}{b}+\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} E\left (\left .\frac {1}{2} i (a+b x)\right |2\right )}{b}\right )\right )}{7 b}\) |
(-2*x*Sech[a + b*x]^(7/2))/(7*b) + (2*((2*Sech[a + b*x]^(5/2)*Sinh[a + b*x ])/(5*b) + (3*(((2*I)*Sqrt[Cosh[a + b*x]]*EllipticE[(I/2)*(a + b*x), 2]*Sq rt[Sech[a + b*x]])/b + (2*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/b))/5))/(7*b)
3.6.36.3.1 Defintions of rubi rules used
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(b*Csc[c + d*x] )^n*Sin[c + d*x]^n Int[1/Sin[c + d*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]
Int[(x_)^(m_.)*Sech[(a_.) + (b_.)*(x_)^(n_.)]^(p_)*Sinh[(a_.) + (b_.)*(x_)^ (n_.)], x_Symbol] :> Simp[(-x^(m - n + 1))*(Sech[a + b*x^n]^(p - 1)/(b*n*(p - 1))), x] + Simp[(m - n + 1)/(b*n*(p - 1)) Int[x^(m - n)*Sech[a + b*x^n ]^(p - 1), x], x] /; FreeQ[{a, b, p}, x] && IntegerQ[n] && GeQ[m - n, 0] && NeQ[p, 1]
\[\int x \operatorname {sech}\left (b x +a \right )^{\frac {9}{2}} \sinh \left (b x +a \right )d x\]
Exception generated. \[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
Timed out. \[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\text {Timed out} \]
\[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \operatorname {sech}\left (b x + a\right )^{\frac {9}{2}} \sinh \left (b x + a\right ) \,d x } \]
\[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\int { x \operatorname {sech}\left (b x + a\right )^{\frac {9}{2}} \sinh \left (b x + a\right ) \,d x } \]
Timed out. \[ \int x \text {sech}^{\frac {9}{2}}(a+b x) \sinh (a+b x) \, dx=\int x\,\mathrm {sinh}\left (a+b\,x\right )\,{\left (\frac {1}{\mathrm {cosh}\left (a+b\,x\right )}\right )}^{9/2} \,d x \]